Finite Group Databasebeta

← #25843

Subgroups — #25843

Order 256.

523 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic
2C2×#499C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic
4#2584C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2059C211yesmaximal, characteristic, complemented
7#2058C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1984C211yesmaximal, characteristic, complemented
11#1864C211yesmaximal, characteristic
12#1864C211yesmaximal, characteristic
13#1862C211yesmaximal, characteristic
14#1862C211yesmaximal, characteristic
15C4×#102C211yesmaximal, characteristic
16#554—24no—
17#554—24no—
18C2×C4⋊Q8C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#125C2211yescharacteristic
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8—21no—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C22×C4⋊C4—21no—
30#499—24no—
31#478—24no—
32C2×#107C2211yescharacteristic
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×C4⋊C8C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#398—21no—
45#395—21no—
46#389—21no—
47#389—21no—
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#389C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8D411yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4⋊Q8—22no—
74C4⋊Q8—22no—
75#125—22no—
76#125—22no—
77#125—22no—
78#125—22no—
79#125—22no—
80#125D414yes—
81C22⋊Q8—28no—
82C22⋊Q8—28no—
83C22⋊Q8—28no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4D414yes—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122#107—22no—
123#107—22no—
124#106—22no—
125#106—22no—
126#106—22no—
127C4⋊C8—22no—
128#102—22no—
129#102—22no—
130#102Q814yescomplemented
131#102Q814yescomplemented
132C4×C8—22no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C2×Q8—24no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—41no—
143C2×Q8—81no—
144C2×Q8—81no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C2×D412yes—
149C2×D4C2×Q812yes—
150C2×D4C2×Q812yes—
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2411yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8C2×Q812yes—
185C2×C8C2×Q812yes—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—22no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4C2×D412yes—
210C4⋊C4C2×D412yes—
211C4⋊C4C2×D412yes—
212C4⋊C4C2×Q812yes—
213C4⋊C4C2×Q812yes—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—81no—
222C42C2×D412yes—
223C42C2×D412yes—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23C22×D411yescharacteristic
232Q8—44no—
233Q8—84no—
234D4—22no—
235D4—22no—
236D4—22no—
237D4C4⋊Q814yes—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—48no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—84no—
276C2×C4—84no—
277C2×C4#12012yes—
278C2×C4#12012yes—
279C2×C4#13512yes—
280C2×C4C22×D411yescharacteristic
281C2×C4C22×D411yescharacteristic
282C2×C4C22×Q811yescharacteristic
283C2×C4C22×Q811yescharacteristic
284C2×C42−1+411yescharacteristic
285C2×C42−1+411yescharacteristic
286C8—22no—
287C8—22no—
288C8—41no—
289C8—41no—
290C22—24no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22C2×#12011yescharacteristic
298C22D4×Q811yescharacteristic
299C22D4×Q811yescharacteristic
300C22C2×#13511yescharacteristic
301C22#57511yescharacteristic
302C22#57611yescharacteristic
303C22#57711yescharacteristic
304C4—22no—
305C4—22no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4C2×C4⋊Q811yescharacteristic
318C4#53611yescharacteristic
319C4D4×Q811yescharacteristic
320C4D4×Q811yescharacteristic
321C2—44no—
322C2#243411yescharacteristic
323C2#300211yescharacteristic
324C2#301211yescharacteristic
325C2#338011yescharacteristic
326C2#338111yescharacteristic
327C2#338811yescharacteristic
328C2#338911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#499C2semidirect
4. #2584C2semidirect
5. #2059C2semidirect
6. #2059C2semidirect
7. #2058C2semidirect
8. #2057C2semidirect
9. #1989C2semidirect
10. #1984C2semidirect
130. #102Q8semidirect
131. #102Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#5542
C2×#4992
C2×#4782
#25842
#20592
#20582
#20572
#19892
#19842
#18642
#18622
C4×#1022